Let $(\Omega,\mu)$ be a $\sigma$-finite measured space. Let $\tau$ be an endomorphism of this space, meaning that $\mu(\tau^{-1}(A))=\mu(A)$ for any $A$. There is a decomposition $\Omega=C \cup D$ such that i) $D$ is a countable union of wandering sets ; ii) any wandering set is a subset of $D$ (mod. null sets) ; iii) one has $C \subset \tau^{-1}(C)$. This decomposition is unique mod. null sets. It is called Hopf decomposition.
I want to understand why it is true in fact that $C=\tau^{-1}(C)$ (mod. null sets). Of course if $\mu(\Omega) < \infty$ this is obvious.
Thank you.